Effectiveness of copula-extreme value theory in estimating value-at-risk: empirical evidence from Asian emerging markets

A traditional Monte Carlo simulation using linear correlations induces estimation bias in measuring portfolio value-at-risk (VaR), due to the well-documented existence of fat-tail, skewness, truncations, and non-linear relations in return distributions. In this paper, we consider the above issues in modeling VaR and evaluate the effectiveness of using copula-extreme-value-based semiparametric approaches. To assess portfolio risk in six Asian markets, we incorporate a combination of extreme value theory (EVT) and various copulas to build joint distributions of returns. A backtesting analysis using a Monte Carlo VaR simulation suggests that the Clayton copula-EVT evinces the best performance regardless of the shapes of the return distributions, and that in general the copulas with the EVT provide better estimations of VaRs than the copulas with conventionally employed empirical distributions. These findings still hold in conditional-coverage-based backtesting. These findings indicate the economic significance of incorporating the down-side shock in risk management.     

价格异常与不对称相关结构下套期保值策略——来自亚洲股指期货市场的实证

在不对称相关结构下,尾部区域期货和现货价格联动模式异常复杂,传统套期保值策略在极端价格行为下将不可避免地出现系统性偏差。为此,恰当的连接函数被引入描述尾部区域的价格联动模式,依据MV框架构建局部套期保值策略,以解决极端价格行为下的套期保值问题。对亚洲市场日经指数和恒生指数期货套期保值的实证研究表明:(1)采用Rotated Gumbel连接函数,在传统MV策略(全局策略)下,虽然在降低套保组合方差方面没有明显优势,但日经指数和恒生指数期货的套期保值成本将显著下降,套保组合收益/组合方差的比率显著上升;(2)如果套期保值者能够对未来形成可靠预期,套期保值的成本将进一步下降,并且现货价格风险也得到有效控制。     

Portfolio selection with commodities under conditional copulas and skew preferences

This article investigates the portfolio selection problem of an investor with three-moment preferences taking positions in commodity futures. To model the asset returns, we propose a conditional asymmetric t copula with skewed and fat-tailed marginal distributions, such that we can capture the impact on optimal portfolios of time-varying moments, state-dependent correlations, and tail and asymmetric dependence. In the empirical application with oil, gold and equity data from 1990 to 2010, the conditional t copulas portfolios achieve better performance than those based on more conventional strategies. The specification of higher moments in the marginal distributions and the type of tail dependence in the copula has significant implications for the out-of-sample portfolio performance.     

Financial crisis,Value-at-Risk forecasts and the puzzle of dependency modeling

Forecasting Value-at-Risk (VaR) for financial portfolios is a crucial task in applied financial risk management. In this paper, we compare VaR forecasts based on different models for return interdependencies: volatility spillover (Engle & Kroner, 1995), dynamic conditional correlations (Engle, 2002, 2009) and (elliptical) copulas (Embrechts et al., 2002). Moreover, competing models for marginal return distributions are applied. In particular, we apply extreme value theory (EVT) models to GARCH-filtered residuals to capture excess returns.Drawing on a sample of daily data covering both calm and turbulent market phases, we analyze portfolios consisting of German Stocks, national indices and FX-rates. VaR forecasts are evaluated using statistical backtesting and Basel II criteria. The extensive empirical application favors the elliptical copula approach combined with extreme value theory (EVT) models for individual returns. 99% VaR forecasts from the EVT-GARCH-copula model clearly outperform estimates from alternative models accounting for dynamic conditional correlations and volatility spillover for all asset classes in times of financial crisis.     

Measuring large comovements in financial markets

A general, copula-based framework for measuring the dependence among financial time series is presented. Particular emphasis is placed on multivariate conditional Spearman's rho (MCS), a new measure of multivariate conditional dependence that describes the association between large or extreme negative returns—so-called tail dependence. We demonstrate that MCS has a number of advantages over conventional measures of tail dependence, both in theory and in practical applications. In the analysis of univariate financial series, data are filtered to remove temporal dependence as a matter of routine. We show that standard filtering procedures may strongly influence the conclusions drawn concerning tail dependence. We give empirical applications to two large data sets of high-frequency asset returns. Our results have immediate implications for portfolio risk management, derivative pricing and portfolio selection. In this context we address portfolio tail diversification and tail hedging. Amongst other aspects, it is shown that the proposed modeling framework improves the estimation of portfolio risk measures such as the value at risk.     

Measuring financial risks with copulas

This paper is concerned with the statistical modeling of the dependence structure of multivariate financial data using the concept of copulas. We select some special copulas and identify the type of dependency captured by each one. We fit copulas to daily returns and simulate from the fitted models. We compare the effect of the choice of copula on risk measures and assess the variability of one-step-ahead predictions of portfolio losses. We analyze extreme scenarios and fit extreme value copulas to the block maxima and minima from daily returns. The stress scenarios constructed are compared to those obtained using models from the extreme value theory. We illustrate the usefulness of the copula approach using two stock market indexes.     

Tails,Fears, and Risk Premia

We show that the compensation for rare events accounts for a large fraction of the average equity and variance risk premia. Exploiting the special structure of the jump tails and the pricing thereof, we identify and estimate a new Investor Fears index. The index reveals large time‐varying compensation for fears of disasters. Our empirical investigations involve new extreme value theory approximations and high‐frequency intraday data for estimating the expected jump tails under the statistical probability measure, and short maturity out‐of‐the‐money options and new model‐free implied variation measures for estimating the corresponding risk‐neutral expectations.     

Asymptotic Analysis of the Loss Given Default in the Presence of Multivariate Regular Variation

Consider a portfolio of n obligors subject to possible default. We propose a new structural model for the loss given default, which takes into account the severity of default. Then we study the tail behavior of the loss given default under the assumption that the losses of the n obligors jointly follow a multivariate regular variation structure. This structure provides an ideal framework for modeling both heavy tails and asymptotic dependence. Multivariate models involving Archimedean copulas and mixtures are revisited. As applications, we derive asymptotic estimates for the value at risk and conditional tail expectation of the loss given default and compare them with the traditional empirical estimates.     

A general approach to integrated risk management with skewed,fat-tailed risks

Integrated risk management for financial institutions requires an approach for aggregating risk types (market, credit, and operational) whose distributional shapes vary considerably. We construct the joint risk distribution for a typical large, internationally active bank using the method of copulas. This technique allows us to incorporate realistic marginal distributions that capture essential empirical features of these risks such as skewness and fat-tails while allowing for a rich dependence structure. We explore the impact of business mix and inter-risk correlations on total risk. We then compare the copula-based method with several conventional approaches to computing risk.     

Managing extreme risks in tranquil and volatile markets using conditional extreme value theory

Financial risk management typically deals with low-probability events in the tails of asset price distributions. To capture the behavior of these tails, one should therefore rely on models that explicitly focus on the tails. Extreme value theory (EVT)-based models do exactly that, and in this paper, we apply both unconditional and conditional EVT models to the management of extreme market risks in stock markets. We find conditional EVT models to give particularly accurate Value-at-Risk (VaR) measures, and a comparison with traditional (Generalized ARCH (GARCH)) approaches to calculate VaR demonstrates EVT as being the superior approach both for standard and more extreme VaR quantiles.     

Jackknife Empirical Likelihood Intervals for Spearman’s Rho

Abstract

In connection with copulas, rank correlation such as Kendall’s tau and Spearman’s rho has been employed in risk management for summarizing dependence between two variables and estimating parameters in bivariate copulas and elliptical models. In this paper a jackknife empirical likelihood method is proposed to construct confidence intervals for Spearman’s rho without estimating the asymptotic variance. A simulation study confirms the advantages of the proposed method.     

Checking for asymmetric default dependence in a credit card portfolio: A copula approach

Traditional credit risk models adopt the linear correlation as a measure of dependence and assume that credit losses are normally-distributed. However some studies have shown that credit losses are seldom normal and the linear correlation does not give accurate assessment for asymmetric data. Therefore it is possible that many credit models tend to misestimate the probability of joint extreme defaults.This paper employs Copula Theory to model the dependence across default rates in a credit card portfolio of a large UK bank and to estimate the likelihood of joint high default rates. Ten copula families are used as candidates to represent the dependence structure. The empirical analysis shows that, when compared to traditional models, estimations based on asymmetric copulas usually yield results closer to the ratio of simultaneous extreme losses observed in the credit card portfolio.Copulas have been applied to evaluate the dependence among corporate debts but this research is the first paper to give evidence of the outperformance of copula estimations in portfolios of consumer loans. Moreover we test some families of copulas that are not typically considered in credit risk studies and find out that three of them are suitable for representing dependence across credit card defaults.     

The choice of the distribution of asset returns: How extreme value theory can help?

One of the issues of risk management is the choice of the distribution of asset returns. Academics and practitioners have assumed for a long time (for more than three decades) that the distribution of asset returns is a Gaussian distribution. Such an assumption has been used in many fields of finance: building optimal portfolio, pricing and hedging derivatives and managing risks. However, real financial data tend to exhibit extreme price changes such as stock market crashes that seem incompatible with the assumption of normality. This article shows how extreme value theory can be useful to know more precisely the characteristics of the distribution of asset returns and finally help to chose a better model by focusing on the tails of the distribution. An empirical analysis using equity data of the US market is provided to illustrate this point.     

Extreme Value Theory and Archimedean Copulas

Using the language of copulas, we generalize the famous Fisher-Tippett Theorem of extreme value theory to the case with sequences of dependent random variables. The dependence structure is modelled using archimedean copulas. This generalization enables to study the behaviour of the maxima of dependent random sequences.     

Selecting copulas for risk management

Copulas offer financial risk managers a powerful tool to model the dependence between the different elements of a portfolio and are preferable to the traditional, correlation-based approach. In this paper, we show the importance of selecting an accurate copula for risk management. We extend standard goodness-of-fit tests to copulas. Contrary to existing, indirect tests, these tests can be applied to any copula of any dimension and are based on a direct comparison of a given copula with observed data. For a portfolio consisting of stocks, bonds and real estate, these tests provide clear evidence in favor of the Student’s t copula, and reject both the correlation-based Gaussian copula and the extreme value-based Gumbel copula. In comparison with the Student’s t copula, we find that the Gaussian copula underestimates the probability of joint extreme downward movements, while the Gumbel copula overestimates this risk. Similarly we establish that the Gaussian copula is too optimistic on diversification benefits, while the Gumbel copula is too pessimistic. Moreover, these differences are significant.     

Minimizing CVaR in global dynamic hedging with transaction costs

This study develops a global derivatives hedging methodology which takes into account the presence of transaction costs. It extends the Hodges and Neuberger [Rev. Futures Markets, 1989, 8, 222–239] framework in two ways. First, to reduce the occurrence of extreme losses, the expected utility is replaced by the conditional Value-at-Risk (CVaR) coherent risk measure as the objective function. Second, the normality assumption for the underlying asset returns is relaxed: general distributions are considered to improve the realism of the model and to be consistent with fat tails observed empirically. Dynamic programming is used to solve the hedging problem. The CVaR minimization objective is shown to be part of a time-consistent framework. Simulations with parameters estimated from the S&P 500 financial time series show the superiority of the proposed hedging method over multiple benchmarks from the literature in terms of tail risk reduction.     

Does the Probability of Informed Trading Model Fit Empirical Data?

The probability of informed trading (PIN) is used widely as a measure of information asymmetry. Relatively little work has appeared on how well PIN models fit empirical trade data. We reveal structural limitations in PIN models by examining their marginal distributions and dependence structures represented by copulas. We develop a distribution-free test of the goodness-of-fit of PIN models. Our results indicate that estimated PIN models have generally poor fit to actual trade data. These results suggest that researchers should be cautious when PIN estimates are plugged into empirical models as explanatory variables.     

Long-term asset tail risks in developed and emerging markets

A power law typically governs the tail decay of financial returns but the constancy of the so-called tail index which dictates the tail decay remains relatively unexplored. We study the finite sample properties of some recently proposed endogenous tests for structural change in the tail index. Given that the finite sample critical values strongly depend on the tail parameters of the return distribution we propose a bootstrap-based version of the structural change test. Our empirical application spans developed and emerging financial asset returns. Somewhat surprisingly, emerging stock market tails are not more inclined to structural change than their developed counterparts. Emerging currency tails, on the contrary, do exhibit structural shifts in contrast to developed currencies. Our results suggest that extreme value theory (EVT) applications in hedging tail risks can assume stationary tail behavior over long time spans provided one considers portfolios that solely consist of stocks or bonds.     

Selective hedging strategies for crude oil futures based on market state expectations

This paper studies the ex-ante selective hedging strategies of crude oil futures contracts based on market state expectations and compares the hedging performances to the traditional minimum variance routine hedging strategies. The main advantage of the proposed method is that it achieves a trade-off between return and risk, rather than hedges risk at all costs. Specifically, we first use a multi-input Hidden Markov Model(HMM) to identify the market state, assess the market’s herding impact, and then integrate the findings of identification and measurement to forecast the price trend. We offer an adjustment criterion for the hedge ratios driven by GARCH²-type models based on the anticipated market state. We conducted an empirical analysis to examine the hedging effect of WTI and Brent crude oil futures, the results indicate that the proposed state-dependent hedging strategies are superior to the traditional model-driven hedging strategies concerning the hedged portfolio based on four criteria. The robustness check reveals that the proposed hedging strategies still outperform in different market situation. The findings can help traders in the crude oil markets, and the methodology can be applied to other energy markets.     

Value-at-risk: Applying the extreme value approach to Asian markets in the recent financial turmoil

Value-at-risk (VaR) measures are generated using extreme value theory by modelling the tails of the return distributions of six Asian financial markets during the recent volatile market conditions. The maxima and minima of these return series were found to be satisfactorily modelled within an extreme value framework and the value at risk measures generated within this structure were found to be different to those generated by variance–covariance and historical methods, particularly for markets characterised by high degrees of leptokurtosis such as Malaysia and Indonesia.